Limit Cycles,Isochronous Centers Of High Dimensional Differential Systems And Traveling Wave Solutions Of Nonlinear Wave Equations

This thesis is devoted to investigate problems on bifurcation of limit cycles and isochronous center conditions on center manifold for higher dimensional systems,and traveling wave solutions of nonlinear wave equations.It is composed of five chapters.In Chapter 1,the historical background and the latest research progress of problems concerned with centers,isochronous centers and bifurcation of limit cycles for higher dimensional polynomial differential systems and traveling wave solutions for nonlinear wave equations are introduced and summarized.Meanwhile,the main work of each Chapter is briefly concluded.In Chapter 2,we deal with the bifurcation problem of limit cycles of positive equilibrium point on the center manifold for a class of three-dimensional cubic Kolmogorov systems.By using the computer algebra software Mathematica and the recursive algorithm for calculating the singular point quantity of three-dimensional polynomial differential systems on the center manifold,the singular point quantities of positive equilibrium point(1,1,1)of the system are calculated,and the necessary and sufficient condition for it to be a 7-order fine focus is obtained.We prove that the positive equilibrium point(1,1,1)of the system on the center manifold can bifurcate 7 small-amplitude limit cycles.In Chapter 3,we give a direct method to study the isochronous center problem of fourdimensional polynomial differential systems on the center manifold.First,the isochronous constants of four-dimensional differential systems are defined and its recursive formulas are given.Then,the necessary conditions of isochronous center can be determined directly through the computation of isochronous constants without calculating the center manifold of four-dimensional systems.With the help of this method,we solve the isochronous center conditions of the origin for a class of four-dimensional quadratic polynomial systems.At the same time,the bifurcation of limit cycles of the system is also studied by using the focus values algorithm of the four-dimensional differential systems on the center manifold.It is proved that there are 5 small-amplitude limit cycles around the origin of the system.In Chapter 4,the small-amplitude solitary periodic wave solutions and the monotonicity of the periodic wave solutions for a class of reaction-diffusion equations were investigated.The technique is based on transforming the reaction-diffusion equation into a ordinary differential equation(traveling wave system)through traveling wave transformation.The focus values and periodic constants of the origin of the corresponding traveling wave system are calculated by recursive algorithm.On this basis,we obtain the necessary and sufficient conditions for the origin to be a 8-order fine focus and center,and prove that 8 small-amplitude limit cycles and at most 3 local bifurcation of critical periods can be bifurcated from the origin of the system,and 3 local bifurcation of critical periods can be reached.Correspondingly,the reaction-diffusion equation has 8 small-amplitude solitary periodic wave solutions bifurcating from the steady-state solution,and the monotonicity of the wavelength function of the periodic wave solution changes 3 times at most.In the last Chapter,the main work of the thesis is summarized,and some prospects for future research work are put forward.